Fixed cost and resource allocation based on DEA cross-efficiency

In many managerial applications, situations frequently occur when a fixed cost is used in constructing the common platform of an organization, and needs to be shared by all related entities, or decision making units (DMUs). It is of vital importance to allocate such a cost across DMUs where there is competition for resources. Data envelopment analysis (DEA) has been successfully used in cost and resource allocation problems. Whether it is a cost or resource allocation issue, one needs to consider both the competitive and cooperative situation existing among DMUs in addition to maintaining or improving efficiency. The current paper uses the cross-efficiency concept in DEA to approach cost and resource allocation problems. Because DEA cross-efficiency uses the concept of peer appraisal, it is a very reasonable and appropriate mechanism for allocating a shared resource/cost. It is shown that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency DEA-based iterative method is further extended into a resource-allocation setting to achieve maximization in the aggregated output change by distributing available resources. Such allocations for fixed costs and resources are more acceptable to the players involved, because the allocation results are jointly determined by all DMUs rather than a specific one. The proposed approaches are demonstrated using an existing data set that has been applied in similar studies.     

Allocating the fixed cost as a complement of other cost inputs: A DEA approach

In cost allocation problem, traditional DEA approaches allocate the fixed cost among a group of decision making units (DMUs), and treat the allocated cost as an extra input of each DMU. If costs except for the fixed cost are regarded as inputs in the cost allocation problem, then it is obvious that the fixed cost is a complement of other inputs rather than an extra independent input. Therefore it is necessary to combine the allocated cost with other cost measures in cost allocation problem. Based on this observation, this paper investigates the relationship between the allocated cost and the DEA efficiency score and develops a DEA-based approach to allocate the fixed cost among various DMUs. An example of allocating advertising expenditure between a car manufacturer and its dealers is presented to illustrate the method proposed in this paper.     

Output–input ratio analysis and DEA frontier

Data envelopment analysis (DEA) is a mathematical programming technique for identifying efficient frontiers for peer decision making units (DMUs). The ability of identifying frontier DMUs prior to the DEA calculation is of extreme importance to an effective and efficient DEA computation. In this paper, we present mathematical properties which characterize the inherent relationships between DEA frontier DMUs and output–input ratios. It is shown that top-ranked performance by ratio analysis is a DEA frontier point. This in turn allows identification of membership of frontier DMUs without solving a DEA program. Such finding is useful in streamlining the solution of DEA.     

Resource allocation based on the DEA model

The purpose of this paper is to develop an approach to a resource-allocation problem that typically appears in organizations with a centralized decision-making environment, for example, police stations, banks, and universities. The central unit is assumed to be interested in maximizing both the total efficiency and the efficiency of the individual unit by allocating available resources to them. Building upon this, we present a data envelopment analysis-based model for allocating input resources to DMUs (the decision-making units) under the framework of multiple objective programming. Numerical examples are used to illustrate the approach.     

Best cooperative partner selection and input resource reallocation using DEA

Reallocation of input resources (RIR) is a process by which certain decision making units (DMUs) reallocate resources among themselves; a process that occurs frequently in many enterprises. In this paper, a new data envelopment analysis (DEA) approach is developed to select the best cooperative partner DMU. Context-dependent DEA is used to identify the different levels of best-practice frontiers. Two DEA-based models are established for two cooperative scenarios, namely, resources pooling only and best-practice sharing. A cooperative method is applied to determine how to reallocate the input resources, and Shapley value is used to estimate the revenue changes that the various DMUs should expect after RIR. Two different situations with different objectives are considered. One objective is to maximize total revenue for the partnership, while the other is to maximize the Shapley value. The proposed approaches are illustrated with two examples.     

Allocating fixed costs and common revenue via data envelopment analysis

An issue of considerable importance involves the allocation of fixed costs or common revenue among a set of competing entities in an equitable way. Based on the data envelopment analysis (DEA) theory, this paper proposes new methods for (i) allocating fixed costs to decision making units (DMUs) and (ii) distributing common revenue among DMUs, in such a way that the relative efficiencies of all DMUs remain unchanged and the allocations should reflect the relative efficiencies and the input-output scales of individual DMUs. To illustrate our methods, numerical results for an example are described in this paper.     

The Adjusted Spherical Frontier Model with weight restrictions

This paper presents the ASFM-lp model, a parametric Data Envelopment Analysis (DEA) model for allocating resources, commonly called inputs. This model considers that a fair allocation of inputs is one that maximizes the DEA-CCR efficiencies of the Decision Making Units (DMUs). The main assumption of the ASFM-lp is the predefined spherical shape of the efficiency frontier. We have demonstrated that our method extends the existing parametric model ASFM to allow the introduction of weight restrictions, which has great importance in practical applications of DEA. Numeric examples are presented to show the application of the method.     

General and multiplicative non-parametric corporate performance models with interval ratio data

The increasing intensity of global competition has led organizations to utilize various types of performance measurement tools for improving the quality of their products and services. Data envelopment analysis (DEA) is a methodology for evaluating and measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. All the data in the conventional DEA with input and/or output ratios assumes the form of crisp numbers. However, the observed values of data in real-world problems are sometimes expressed as interval ratios. In this paper, we propose two new models: general and multiplicative non-parametric ratio models for DEA problems with interval data. The contributions of this paper are fourfold: (1) we consider input and output data expressed as interval ratios in DEA; (2) we address the gap in DEA literature for problems not suitable or difficult to model with crisp values; (3) we propose two new DEA models for evaluating the relative efficiencies of DMUs with interval ratios, and (4) we present a case study involving 20 banks with three interval ratios to demonstrate the applicability and efficacy of the proposed models where the traditional indicators are mostly financial ratios.     

The interval efficiency based on the optimistic and pessimistic points of view

Data envelopment analysis (DEA) is a data-oriented approach for evaluating the performances of a set of peer entities called decision-making units (DMUs), whose performance is determined based on multiple measures. The traditional DEA, which is based on the concept of efficiency frontier (output frontier), determines the best efficiency score that can be assigned to each DMU. Based on these scores, DMUs are classified into DEA-efficient (optimistic efficient) or DEA-non-efficient (optimistic non-efficient) units, and the DEA-efficient DMUs determine the efficiency frontier. There is a comparable approach which uses the concept of inefficiency frontier (input frontier) for determining the worst relative efficiency score that can be assigned to each DMU. DMUs on the inefficiency frontier are specified as DEA-inefficient or pessimistic inefficient, and those that do not lie on the inefficient frontier, are declared to be DEA-non-inefficient or pessimistic non-inefficient. In this paper, we argue that both relative efficiencies should be considered simultaneously, and any approach that considers only one of them will be biased. For measuring the overall performance of the DMUs, we propose to integrate both efficiencies in the form of an interval, and we call the proposed DEA models for efficiency measurement the bounded DEA models. In this way, the efficiency interval provides the decision maker with all the possible values of efficiency, which reflect various perspectives. A numerical example is presented to illustrate the application of the proposed DEA models.     

Centralized DEA models with the possibility of downsizing

While traditional data envelopment analysis (DEA) models assess the relative efficiency of similar, independent decision making units (DMUs) centralized DEA models aim at reallocating inputs and outputs among the units setting new input and output targets for each one. This system point of view is appropriate when the DMUs belong to a common organization that allocates their inputs and appropriates their outputs. This intraorganizational perspective opens up the possibility that greater technical efficiency for the organization as a whole might be achieved by closing down some of the existing DMUs. In this paper, we present three centralized DEA models that take advantage of this possibility. Although these models involve some binary variables, we present efficient solution approaches based on Linear Programming. We also present some numerical results of the proposed models for a small problem from the literature.     

Efficiency evaluation model with constraint resource: an application to banking operations

Competition is often presented in a free market. Efficiency evaluation of decision-making units (DMUs) needs accommodation of such competition among various units due to constrained resources. This paper develops an innovative quantitative approach to address the above-mentioned performance evaluation problem with constrained resource using output-oriented DEA. The proposed model allows DMUs to identify the maximum input reduction and resource savings to achieve performance improvement. Relations between the proposed model and classical output-oriented DEA models are explored and some economic insights are derived from these models. The proposed approach is validated by use of computational examples.     

DEA model with shared resources and efficiency decomposition

Data envelopment analysis (DEA) has proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, DMUs have a two-stage network process with shared input resources used in both stages of operations. For example, in hospital operations, some of the input resources such as equipment, personnel, and information technology are used in the first stage to generate medical record to track treatments, tests, drug dosages, and costs. The same set of resources used by first stage activities are used to generate the second-stage patient services. Patient services also use the services generated by the first stage operations of housekeeping, medical records, and laundry. These DMUs have not only inputs and outputs, but also intermediate measures that exist in-between the two-stage operations. The distinguishing characteristic is that some of the inputs to the first stage are shared by both the first and second stage, but some of the shared inputs cannot be conveniently split up and allocated to the operations of the two stages. Recognizing this distinction is critical for these types of DEA applications because measuring the efficiency of the production for first-stage outputs can be misleading and can understate the efficiency if DEA fails to consider that some of the inputs generate other second-stage outputs. The current paper develops a set of DEA models for measuring the performance of two-stage network processes with non splittable shared inputs. An additive efficiency decomposition for the two-stage network process is presented. The models are developed under the assumption of variable returns to scale (VRS), but can be readily applied under the assumption of constant returns to scale (CRS). An application is provided.     

Centralized resource allocation with stochastic data

Data Envelopment Analysis (DEA) is a technique based on mathematical programming for evaluating the efficiency of homogeneous Decision Making Units (DMUs). In this technique inefficient DMUs are projected on to the frontier which constructed by the best performers. Centralized Resource Allocation (CRA) is a method in which all DMUs are projected on to the efficient frontier through solving just one DEA model. The intent of this paper is to present the Stochastic Centralized Resource Allocation (SCRA) in order to allocate centralized resources where inputs and outputs are stochastic. The concept discussed throughout this paper is illustrated using the aforementioned example.     

In the determination of weight sets to compute cross-efficiency ratios in DEA

Data envelopment analysis (DEA) measures the production performance of decision-making units (DMUs) which consume multiple inputs and produce multiple outputs. Although DEA has become a very popular method of performance measure, it still suffers from some shortcomings. For instance, one of its drawbacks is that multiple solutions exist in the linear programming solutions of efficient DMUs. The obtained weight set is just one of the many optimal weight sets that are available. Then why use this weight set instead of the others especially when this weight set is used for cross-evaluation? Another weakness of DEA is that extremely diverse or unusual values of some input or output weights might be obtained for DMUs under assessment. Zero input and output weights are not uncommon in DEA. The main objective of this paper is to develop a new methodology which applies discriminant analysis, super-efficiency DEA model and mixed-integer linear programming to choose suitable weight sets to be used in computing cross-evaluation. An advantage of this new method is that each obtained weight set can reflect the relative strengths of the efficient DMU under consideration. Moreover, the method also attempts to preserve the original classificatory result of DEA, and in addition this method produces much less zero weights than DEA in our computational results.     

Finding common weights based on the DM's preference information

Data Envelopment Analysis (DEA) is basically a linear programming-based technique used for measuring the relative performance of organizational units, referred to as Decision Making Units (DMUs). The flexibility in selecting the weights in standard DEA models deters the comparison among DMUs on a common base. Moreover, these weights are not suitable to measure the preferences of a decision maker (DM). For dealing with the first difficulty, the concept of common weights was proposed in the DEA literature. But, none of the common weights approaches address the second difficulty. This paper proposes an alternative approach that we term as ‘preference common weights’, which is both practical and intellectually consistent with the DEA philosophy. To do this, we introduce a multiple objective linear programming model in which objective functions are input/output variables subject to the constraints similar to the equations that define production possibility set of standard DEA models. Then by using the Zionts–Wallenius method, we can generate common weights as the DM's underlying value structure about objective functions.     

Spherical frontier DEA model based on a constant sum of inputs

In this paper, a Data Envelopment Analysis (DEA) model in which a fixed input needs to be assigned to a group of Decision-Making Units (DMUs) is presented. This is performed by assuming the existence of a geometric place represented by a sphere that characterizes the DEA frontier. It is shown that, under this assumption, it becomes relatively easy to find a way to distribute the fixed input to all DMUs, by considering that the individual assignments will be fair through the requirement that all DMUs be efficient or, in other words, be located on the spherically shaped efficiency frontier. A model is presented and results are compared to those obtained by using two different methods proposed in the literature within the same context.     

基于复合DEA模型高校办学效益评价方法研究

传统DEA只能在固定投入(或产出)的情况下,将产出(或投入)尽量扩大(或缩小),而造成决策单元非DEA有效的原因,既有投入方面的因素,也有产出方面的因素,是由投入和产出两方面的原因共同造成的,而不仅仅是一方面的原因.本文考虑投入和产出两方面的因素,构造了复合DEA模型,并研究了基于复合DEA模型高校办学效益评价方法.     

Random effects logistic regression model for ranking efficiency in data envelopment analysis

Ranking efficiency based on data envelopment analysis (DEA) results can be used for grouping decision-making units (DMUs). The resulting group membership can be partly related to the environmental characteristics of DMU, which are not used either as input or output. Utilizing the expert knowledge on super efficiency DEA results, we propose a multinomial Dirichlet regression model, which can be used for the purpose of selection of new projects. A case study is presented in the context of ranking analysis of new information technology commercialization projects. It is expected that our proposed approach can complement the DEA ranking results with environmental factors and at the same time it facilitates the prediction of efficiency of new DMUs with only given environmental characteristics.     

A new robust DEA model and super-efficiency measure

Applications of traditional data envelopments analysis (DEA) models require knowledge of crisp input and output data. However, the real-world problems often deal with imprecise or ambiguous data. In this paper, the problem of considering uncertainty in the equality constraints is analyzed and by using the equivalent form of CCR model, a suitable robust DEA model is derived in order to analyze the efficiency of decision-making units (DMUs) under the assumption of uncertainty in both input and output spaces. The new model based on the robust optimization approach is suggested. Using the proposed model, it is possible to evaluate the efficiency of the DMUs in the presence of uncertainty in a fewer steps compared to other models. In addition, using the new proposed robust DEA model and envelopment form of CCR model, two linear robust super-efficiency models for complete ranking of DMUs are proposed. Two different case studies of different contexts are taken as numerical examples in order to compare the proposed model with other approaches. The examples also illustrate various possible applications of new models.     

DEA models for minimizing weight disparity in cross-efficiency evaluation

Cross-efficiency evaluation is a commonly used approach for ranking decision-making units (DMUs) in data envelopment analysis (DEA). The weights used in the cross-efficiency evaluation may sometimes differ significantly among the inputs and outputs. This paper proposes some alternative DEA models to minimize the virtual disparity in the cross-efficiency evaluation. The proposed DEA models determine the input and output weights of each DMU in a neutral way without being aggressive or benevolent to the other DMUs. Numerical examples are tested to show the validity and effectiveness of the proposed DEA models and illustrate their significant role in reducing the number of zero weights.